X(92) = X(92) CEVAPOINT OF INCENTER AND CLAWSON POINT

Trilinears

\(csc 2A : csc 2B : csc 2C\)

\(cot A + tan A : :\)

Barycentrics

\(sec A : sec B : sec C\)

Notes

Let LA be the line through X(4) parallel to the internal bisector of angle A, and let A’ = BC∩LA. Define B’ and C’ cyclically.

Alexei Myakishev, “The M-Configuration of a Triangle,” Forum Geometricorum 3 (2003) 135-144,

proves that the lines AA’, BB’, CC’ concur in X(92). He notes that another construction follows from Proposition 2 of the article: let A1 be the midpoint of the arc BC of the circumcircle that passes through A, and let A2 be the point, other than A, in which the A-altitude meets the circumcircle. Let A″ = A1A2∩BC. Define B″ and C″ cyclically. Then the lines AA″, BB″, CC″ concur in X(92).

Suppose that T = A’B’C’ is a central triangle. Let A’’ be the pole with respect to the polar circle of the line B’C’, and define B’’ and C’’ cyclically. The appearance of T in the following list means that the lines AA’’, BB’’, CC’’ concur in X(92): Feurerbach, incentral, excentral, extangents, Apollonius, mixtilinear excentral. (Randy Hutson, December 26, 2015)

X(92) lies on these lines: 1,29 2,273 4,8 7,189 10,1838 19,27 25,242 28,2975 31,162 33,1897 34,1220 38,240 40,412 47,91 48,2167 53,4415 55,243 56,1940 57,653 81,2995 85,331 100,917 108,1311 171,1430 226,342 239,607 255,1087 257,297 264,306 304,561 345,3262 388,1118 394,1943 406,1068 427,2969 429,3948 429,3948 459,1446 497,1857 518,1859 608,894 651,2988 823,2349 938,3176 942,1148 960,1882 984,1860 994,1845 1146,1952 1172,2997 1211,1865 1309,2717 1435,3306 1585,1659 1621,4183 1707,1733 1726,1746 1731,1751 1785,4656 1842,1891 1844,3874 1870,5136 1947,2994 1954,1955 1956,2632 1973,3112 2064,3596 2331,5256 2399,4391 3064,4468 4198,4968

X(92) = isogonal conjugate of X(48)

X(92) = isotomic conjugate of X(63)

X(92) = anticomplement of X(1214)

X(92) = anticomplementary conjugate of X(2897)

X(92) = Fuhrmann-circle-inverse of X(5174)

X(92) = X(i)-Ceva conjugate of X(j) for these (i,j): (85, 342), (264,318), (286,4), (331,273)

X(92) = cevapoint of X(i) and X(j) for these (i,j): (1,19), (4,281), (47,48), (196,278)

X(92) = X(i)-cross conjugate of X(j) for these (i,j): (1,75), (4,273), (19,158), (48,91), (226,2), (281,318)

X(92) = crosspoint of X(i) and X(j) for these (i,j): (85,309), (264,331)

X(92) = crossdifference of every pair of points on line X(810)X(822)

X(92) = X(275)-aleph conjugate of X(47)

X(92) = X(i)-beth conjugate of X(j) for these (i,j): (92,278), (312,329), (648,57)

X(92) = {X(19),:ref:X(63) <X(63)>}-harmonic conjugate of X(1748)

X(92) = barycentric product of PU(20)

X(92) = trilinear product of PU(i) for these i: 21, 45

X(92) = bicentric sum of PU(130)

X(92) = midpoint of PU(130)

X(92) = trilinear product X(2)

X(92) = trilinear pole of line X(240)X(522) (polar of X(1) wrt polar circle)

X(92) = pole of antiorthic axis wrt polar circle

X(92) = X(6)-isoconjugate of X(3)

X(92) = X(48)-isoconjugate (polar conjugate) of X(1)

X(92) = X(88)-isoconjugate of X(23202)

X(92) = X(91)-isoconjugate of X(563)

X(92) = perspector of ABC and extraversion triangle of X(92) (which is also the anticevian triangle of X(92))

X(92) = crosspoint of X(1) and X(19) wrt excentral triangle

X(92) = crosspoint of X(47) and X(48) wrt excentral triangle

X(92) = perspector of ABC and cross-triangle of Gemini triangles 37 and 38

X(92) = perspector of ABC and cross-triangle of ABC and Gemini triangle 37

X(92) = perspector of ABC and cross-triangle of ABC and Gemini triangle 38

X(92) = barycentric product of vertices of Gemini triangle 37

X(92) = barycentric product of vertices of Gemini triangle 38